What is the limiting factor in generating strong electromagnets?

Since magnetic field strength only depends on currect, it seems like you could generate a very powerful magnet simply by connecting a 1 volt battery to a low resistance coil. The wire's resistance seems like it would limit the maximum current to some degree, but is that all there is to it? It doesn't seem plausible to me that a magnetic field generated by connecting a 1 volt battery to a superconducting material could exert a high Lorenz Force on anything.

What counts for strength of an electromagnet is the current times the number of turns. You can use thinner wire to get more turns, but then it will have a lot of resistance and will heat up. Thus the key to a stron magnet is low resistivity of your wire and good way of keeping it cool so the isolation doesn´t break down. See for example this video of how a super strong electromagnet is constructed

You also get more useful magnetism when both N and S poles are on a single side. Like in a horse shoe or "pot" magnet. The lifting magnet in a junk yard or a magnetic grinding chuck( work holding) are made like a "pot" magnet.

There are other ways to increase the density of a magnetic field. One can decrease the dipole area. Magnetic lines are not self-repulsive like electric lines. Magnetic fields can be converged much greater (in a much smaller area) than the electric and therefore much more power density.

And if one could in our present day circuits, increasing the velocity of the current, (not the amount), would increase the density.

As we learn to apply super conductivity, we should see some powerful magnetic fields.

Since magnetic field strength only depends on currect, it seems like you could generate a very powerful magnet simply by connecting a 1 volt battery to a low resistance coil. The wire's resistance seems like it would limit the maximum current to some degree, but is that all there is to it? It doesn't seem plausible to me that a magnetic field generated by connecting a 1 volt battery to a superconducting material could exert a high Lorenz Force on anything.

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A real battery is actually actually an "ideal" battery with a resistor in series (known as "internal resistance"). Therefore the max current is the voltage of the ideal battery divided by the internal resistance.

That max current, is only possible for a short time because the battery will likely overheat and pop as the electrolyte vaporizes.

The max current is also influenced by the resistance of the copper. As mentioned above, more turns means stronger field but it is also a longer piece of copper and thinner than one would use for fewer turns with a larger diameter. It all comes down to some calculations that may be already done for you with some type of inductor calculator.